Volume preserving mean curvature flows near strictly stable sets in flat torus

Abstract

In this paper we establish a new stability result for smooth volume preserving mean curvature flows in flat torus T n in dimensions n = 3, 4. The result says roughly that if an initial set is near to a strictly stable set in T n in H 3 -sense, then the corresponding flow has infinite lifetime and converges exponentially fast to a translate of the strictly stable (critical) set in W 2,5 -sense.